I have a large vector .eps file of 8 logo samples. At 100% zoom, it is crisp and gorgeous, about 4 inches wide and 1.5 inches high on my laptop screen. How do I export it at that size and save it in the same quality, but png format at the same size? I'm new to Illustrator and it is different from Photoshop function wise.

You can't get the "same quality" when comparing vector(ai) to raster (png), Vector is a resolution independent format and will always result in better quality when used. Raster content is constructed of pixels and will always be bound by pixel limitations. What you can do is get a good quality raster image from a vector image.

Open the file in Illustrator.
Choose File > Save For Web
The Save For Web dialog window will open
Click the "Image Size" tab on the right of the window.
Input the dimensions you want.
Under Anti-aliasing, choose "Art Optimized" in most cases
Click Save.

This will save a web-resolution png file of the contents.

Save for web works prety much the same in Illustrator as it does in Photoshop.